A random CLT for dependent random variables

We introduce a new condition for {Y[tau]n} to have the same asymptotic distribution that {Yn} has, where {Yn} is a sequence of random elements of a metric space (S, d) and {[tau]n} is a sequence of random indices. The condition on {Yn} is that maxi[set membership, variant]Dnd(Yi, Yan)-->p0 as n --> [infinity], where Dn = {i: ki-kan <= [delta]ankan} and {[delta]n} is a nonincreasing sequence of positive numbers. The condition on {[tau]n} is that P((k[tau]n/kan)-1 > [delta]an) --> 0 as n --> [infinity]. Under these conditions, we will show that d(Y[tau]n, Yan) --> P0 and apply this result to the CLT for a general class of sequences of dependent random variables.